January 22, 2016

I sing of prows and bows, bulbous and otherwise. It seems all  of the ships (cruise ships, bulk freighters, car carriers) entering harbor have bulbous bows.  I wonder why. I suspect the answer has to do with the Reynolds Number, and maybe also the Froude Number.

The Reynolds and Froude Numbers are examples of ratios that show up again and again in various calculations, so much so they are accorded names. Perhaps the best known ratio is the Mach Number, which is simply the ratio of the speed of an object divided by the speed of sound in the fluid the object is in. Since it is a speed divided by a speed, the units cancel out and the Mach Number is dimensionless, a number with no units, i.e., feet, seconds, etc. In fact, all of these ratios are dimensionless.

Consider United Airlines flight 144, a Boeing 757-200 flying from Chicago to San Francisco, at the normal cruising altitude of 36,000 ft. The air traffic controller in Denver wants to know the aircraft’s speed, to make sure the planes in that corridor remain properly spaced. She, in the terse style of air traffic controllers the world over, makes this transmission:

               United 144, Denver Center: State your Mach.

The United flight crew dutifully responds:

               Denver Center, United 144: Mach is zero point eight two.

That is all she needs to know. She asks the same question of the Delta Boeing 737-800 ahead of United 144, and the JetBlue Airbus A321 trailing, and she has the information she needs.

You might ask yourself, yes, but how fast is that United flight going? Recall the definition of the Mach Number:

We need to know the speed of sound in air at 36,000 feet. This is a bit tricky – the speed of sound in air is not constant; it varies with temperature and pressure. A little research finds an appropriate equation

where A is altitude in 1000 feet. United 144 is cruising at 36,000 feet, so A is 36, and we can solve for the speed of sound in air at 36,000 feet, which is 574 knots. Since United 144 told us it was traveling at Mach 0.82, we have

and so United 144 is traveling at 471 knots, or 535 miles per hour.

Let’s say you are driving 45 in an area with a 30 mile per hour speed limit. A cop pulls you over, and asks if you know how fast you were going. Here’s what you do: Express your speed as a Mach Number. Forty five miles per hour is 66 feet per second, and the speed of sound at sea level is about 1100 feet per second, so

and you can tell the cop you were going at Mach 0.006. This sounds a whole lot slower than 45 and just might save you a ticket, points, and a fine.

By the way, the Mach Number is named for the efforts of Ernst Mach, an Austrian physicist who, in the 1870s and 1880s, studied the acoustics of supersonic projectiles, among other things.

The Reynolds Number is like the Mach Number in that it is a dimensionless ratio of two things, but the things are a little more difficult to envision. The Reynolds Number is defined as

where the inertial forces are the forces causing an object to move through a fluid, and the viscous forces are the drag and friction forces on the object generated by the interaction of the object with the fluid. Another way to state the Reynolds Number is

where the characteristic length and velocity are dictated by the problem being studied.

Let’s assume you’re a passenger on United 144, cruising at 36,000 feet, and you decide to open the emergency exit and jump out. You tuck yourself into a ball and begin your descent. At first, under the influence of gravity, you accelerate, but as you go faster, the drag forces increase. The drag forces soon equal the gravitational (inertial) forces, and you reach a constant velocity of about 140 miles per hour. In this case, the characteristic length would be the diameter of the ball you’ve formed, about three feet, and the characteristic velocity would be what is called the terminal settling velocity, about 140 miles per hour. Your Reynolds Number would be

The viscosity of the atmosphere at -40 C is about 0.00011 square feet per second, so, after converting miles per hour to feet per second, you calculate

You could calculate your Mach Number as well. By use of an earlier equation, we can calculate the speed of sound in air at, say, 10,000 feet. which comes out to about 640 knots, or about 725 miles per hour.

Your Mach Number is therefore

An engineer, looking at the Reynolds Number, would say you were experiencing fully developed turbulent flow. Looking at the Mach Number, she would say you were going pretty damn fast. You’d probably be thinking other things.

I wonder what Francoise de Moriere was thinking on the evening of October 19, 1962. Francoise was a stewardess, as they were called then, on an Allegheny Airlines Convair 440, and she fell to her death when an emergency door inexplicably opened as the plane was at 1,500 feet on final approach to Bradley Airport, near Hartford, Connecticut. James Dickey wondered the same thing. He later wrote, of his poem Falling:

The original idea came out of a newspaper item I once read to the effect that an Allegheny Airlines stewardess had fallen out of an airplane and  was found later on, dead. But when you have a little hint like this that entertains your imagination, you take off with it and make your own thing out of it.

Dickey changed the location to Kansas; the stewardess fell towards the rich black earth of the Corn Belt. The poem begins with the stewardess being sucked out of the plane.

                                                                           As though she blew
The door down with a silent blast from her lungs    frozen    she is black
Out finding herself with the plane nowhere and her body taken by the throat

She thinks of herself in several ways as she falls – an owl looking for chickens, a goddess, a diver. She imagines herself diving into a pond:

                                                                                                if she fell
Into water she might live    like a diver.    Cleaving.…perfect    plunge
Into another    heavy silver     unbreathable   slowing    saving
Element: there is water    there is time to perfect all the fine
Points of diving    feet together    toes pointed     hands shaped right
To insert her into water like a needle     to come out healthily dripping
And be handed a Coca-Cola

Knowing she will be considered special, she undresses as she falls.

 

                                                                                              there is no
Way to back off    from her chosen ground     but she sheds the jacket
With its silver sad impotent wings    sheds the bat’s guiding tailpiece
Of her skirt    the lightning-charged clinging of her blouse    the intimate
Inner flying-garment of her slip in which she rides like the holy ghost
Of a virgin    sheds the long windsocks of her stockings     absurd
Brassiere    then feels the girdle required by regulations squirming
Off her: no longer monobuttocked    she feels the girdle flutter     shake
In her hand    and float     upward     her clothes rising off her ascending
Into cloud    and fights away from her head the last sharp dangerous shoe
Like a dumb bird    and now will drop in    soon   now will drop
In like this

the greatest thing that ever came to Kansas

Joyce Carol Oates called Falling “an astonishing poetic feat.” I’ve always been intrigued by her, not least because many of her books are set in upstate New York. I recently read and enjoyed Carthage.  Perhaps she will be the topic of one of these letters.

James Dickey either knew nothing of Reynolds and Mach Numbers, or chose not to include them in Falling. I’m glad – I like it the way it is. In fact, he said of the poem

I felt justified in writing “Falling” the way I did. I wouldn’t want to go back and try to write it again. I suppose there are faults in it which people will be pointing out to me for years, but I did it the way I wanted to do it, and I’ll stand by that.

The Reynolds Number did not appear in Falling, but it does show up in other strange places. Here’s one you may not have thought of. In 1997, NASA launched the Cassini-Huygens mission to Saturn and Titan, Saturn’s largest moon. The Cassini attained orbit around Saturn in 2004, and on Christmas Day, on a close approach to Titan, the Huygens probe separated, and, after a bit of maneuvering, parachuted through Titan’s atmosphere and landed on her surface, sending back data and images as it fell and for about 90 minutes after landing. The images caused great excitement.

The surface conditions on Titan are such that the hydrocarbons – methane (CH₄), ethane (C₂H₆), and maybe propane (C₃H₈) can exist as solid, liquid and vapor, just like water on Earth. There might therefore be a hydrocarbon cycle on Titan, with hydrocarbon clouds and rain, rivers and lakes. In fact, radar returns from Cassini showed large flat areas on Titan thought to be hydrocarbon lakes. Can we make some reasonable guesses about flows of liquid hydrocarbons in the channels in the images?

On Earth, the velocity of water flowing in a river channel is reasonably well understood, and depends on three things:

This makes sense, if you think about it. The channel slope adds gravitational energy to the flow, forcing the water downstream, and the friction forces act to retard the flow. The result is usually a nearly constant flow velocity in a given channel geometry. In this sense, the fluid flow velocity problem is similar to jumping out of an airplane and reaching your terminal settling velocity – in both cases, the friction (drag) forces equal the gravitational (inertial) forces.

Mission scientists know about the optics on the Huygens probe, and so with a little bit of work one can estimate the width of a typical section of channel to be about 30 meters, or close to 100 feet. In the absence of other information, assume the channel is rectangular, with vertical sides. We don’t have any direct data for channel slope, but the general topography is pretty rugged, so try slopes of 1, 2, 3, 4 and 5 %.

Acceleration due to gravity on Titan is 1.35  meters per second squared, or about 14% that of Earth. The surface temperature is about -180 C; atmospheric pressure (like Earth, Titan’s atmosphere is mostly nitrogen) about 145 kilopascals, as compared to about 100 kilopascals on Earth.

We can now use an equation to calculate flow velocity. Based on knowledge of water flow on Earth, there are several equations to choose from. We used this one:

where g is Titan’s gravity, f a friction factor, D a number dependent on channel geometry, and sinS is the sine of the slope, i.e. 0.01, 0.02. We know the value of g and have made reasonable assumptions about channel geometry (rectangular) and slope. The only thing left is f.

We used this rather imposing looking equation to solve for the friction factor f:

where k_s is based on the size of the pebbles, stones, rocks in the channel, D is the same as the equation above, and Re is our old friend the Reynolds Number.

The image to the left is from Titan’s surface, taken just after the Huygens lander touched down. We used this image to estimate the size of the rocks in view and hence k_s. For the Reynolds Number, we used the viscosity of liquid methane at -180 C, a characteristic length based on the assumed channel geometry, and an assumed flow velocity V.

 

We solve for the final flow velocity by an iterative process. It turns out a reasonable estimate for the velocity of liquid methane flowing through a channel on the surface of Titan is about 0.8 meters per second, or about 2.5 feet per second. By the way, this is a typical value for the velocity of water in a stream on Earth.

I think this similarity of velocities should be celebrated, and hereby propose the following ditty:

On Earth and TItan
Fluids flow, aliken.

OK, so I’m not James Dickey, but at least my ditty has some truth to it unlike, for example

A pints a pound
The world around.

which is false for both US (1.04 lbs) and Imperial (1.25 lbs) pints, at least when the pints are filled with beer.

By the way, the Cassini is still orbiting around Saturn, and occasionally gets close enough to Titan to gather more information. Here is a 2006 false color image derived from the Cassini’s radar system. The dark regions are areas of low radar backscatter, interpreted as  hydrocarbon lakes.

Perhaps, even as you’re reading this, the Sirens of Titan are on an excursion, in rowboats on Ligea Mare, with brightly colored parasols as protection from hydrocarbon droplets falling from the Titian sky. Perhaps their rowboats have bulbous bows. If so, the Sirens surely used the idea of the Reynolds Number in designing them. Of course, the Sirens would not have known that, on Earth, the number is named for Osborne Reynolds, an English mechanical engineer. Maybe the Sirens named the ratio after one of their eminent engineers. Maybe on Titan it is known as the Potrezebie Number.

I don’t know how the Sirens designed their bulbous bows. Here on Earth, it is pretty tricky, and there is not yet, in so far as I can discern, a simple design method. Rather, ship designers resort to scale models, and test the models in tanks.

The trick here is to run the model so that the Reynolds Number, and hence flow regime the model experiences, is the same as the full scale ship will experience. Different model configurations are tested, and the one that uses the least energy to maintain a constant speed is deemed the most efficient.

Bulbous bows work by reducing drag, and the results are quite remarkable, reducing fuel usage by 12 – 15 % in large vessels with them as compared to similar sized vessels without them. They work best when the ship maintains a constant speed, and for long distances. That’s why they are on cruise ships, bulk freighters, and tankers, but not, most likely, rowboats plying Titan’s hydrocarbon waterways.

You may be wondering why and how I developed an interest in bulbous bows. When I was 10 or 11 I came across a copy of Playboy that featured photos of the very buxom British model June Wilkinson. (Google her name – I bet you’ll see what I mean). One particularly memorable photo had her leaning back, with a full glass of champagne balanced on each of her very large and otherwise unadorned breasts. The caption for that, or perhaps another, photo went something like: ‘June, proud of her prow, . . .’

As you can surmise, this made quite an impression on me. I’ve been interested in prows and bows, bulbous and otherwise, ever since.

 

Notes, sources, credits:

The full text of Falling can be found at http://www.poetryfoundation.org/poem/.

The quotes from James Dickey and Joyce Carol Oates are from http://www.english.illinois.edu/maps/poets/a_f/dickey/falling.htm

The two Titan images are copyright  ESA/NASA/JPL/University of Arizona. I downloaded them from http://www.esa.int/spaceinimages/ which has many more  images from Huygens and other space probes.

The Cassini radar image is copyright NASA/JPL/USGS. See http://saturn.jpl.nasa.gov/  for more information about the Cassini-Huygens mission.

I assigned the problem of methane flow on Titan to five students in my senior course in Open Channel Hydraulics. The five – Mabel Gutliph, Matt Huchzermeier, Tyler Kreider, Jeff Newsome, Trevor Schlossnagle – attacked the problem with skill and enthusiasm. I shared their results with researchers at MIT, who were as impressed with their work as I was.